0021d: Article 14 (Euler Four-Square Variants)

Abstract:An infinite family of 8-square identities (which can be specialized into a 4-square) will be given.

I. Introduction

II. A Family of 4-Square Identities

III. A Family of 8-Square Identities

I. Introduction

In the previous article, Pfister’s 16-Square Identity, aside from giving a non-bilinear 16-square identity, a variation of Euler’s Four-Square Identity with only two bilinear zi was also mentioned, namely,

One can ask a reasonable question: How many four-square identities are there?It can be shown that there are an infinite number of four-square and eight-square identities where not all zi are bilinear.

II. A Family of 4-Square Identities

In contrast to the Pfister 4-square, the one below generally has only one bilinear zi,

for arbitrary n.Of course, for n = x4, then k = 0, all the zi become bilinear, hence the Euler 4-square can be considered a special case of this family.The {ai, bi, ci} also satisfy the nice relations,

a12 + b12 + c12 = x12 + x22 + x42

a22 + b22 + c22 = x12 + x32 + x42

a32 + b32 + c32 = x22 + x32 + x42

and,

a12 + a22 + a32= x22 + x32 + x42

b12 + b22 + b32 = x12 + x32 + x42

c12 + c22 + c32= x12 + x22 + x42

If we include the leading coefficients of the first three zias {a0, b0, c0} = {x1, x2, x3}, then the last three relations simplify as,

Σ ai2 = Σ bi2 = Σ ci2 = Σ xj2

However, this 4-square family in turn is just a special case of an 8-square family.

III. A Family of 8-Square Identities

An 8-square identity with four bilinear zi was given in Pfister’s 16-Square Identity. But we can also give one with five bilinear zi.First, define the following variables,

where,

m = x4-n

and the other variables to be defined later. Also,

For the latter set, note that, for any value of the variables, then,

hence, they are just three versions of the Euler 4-Square. And where do the {ai, bi, ci} appear?In an 8-square identity, namely,

and we can now define {pi, qi, ri, si, t} in terms of the xi and n as,

for t = 0, and arbitrary n.(The variable t was just a place-holder.)With these definitions, it can be seen that,

Again, if we include the leading coefficients as {a0, b0, c0} = {x1, x2, x3}, then the last three relations simplify as,

Σ ai2 = Σ bi2 = Σ ci2 = Σ xk2

While n is arbitrary, there is no value n such that all the zi become bilinear, hence this 8-square family cannot reduce to the Degen-Graves 8-square. However, if we set {xi, yi} = 0 for all i > 4, then it does reduce to the 4-square family in the previous section where, as was pointed out already, the special case n = x4 further reduces it to the Euler 4-square. There is in fact an 8-square family that can reduce to the Degen-Graves, but its variables are too tediously complicated for this article.The one given suffices to prove there is an infinite number of 8-square identities.